Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393881 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(2,4)(6,8);(10,14)(13,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264835, #72264841, #72264845, … |
| maximal normal subgroup of | #72264835, #72264841, #72264845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312369);